Journal of Applied Probability

Estimation of integrals with respect to infinite measures using regenerative sequences

Krishna B. Athreya and Vivekananda Roy

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


Let f be an integrable function on an infinite measure space (S, S, π). We show that if a regenerative sequence {Xn}n≥0 with canonical measure π could be generated then a consistent estimator of λ ≡ ∫Sfdπ can be produced. We further show that under appropriate second moment conditions, a confidence interval for λ can also be derived. This is illustrated with estimating countable sums and integrals with respect to absolutely continuous measures on Rd using a simple symmetric random walk on Z.

Article information

J. Appl. Probab., Volume 52, Number 4 (2015), 1133-1145.

First available in Project Euclid: 22 December 2015

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 65C05: Monte Carlo methods
Secondary: 60F05: Central limit and other weak theorems

Markov chain Monte Carlo improper target random walk regenerative sequence


Athreya, Krishna B.; Roy, Vivekananda. Estimation of integrals with respect to infinite measures using regenerative sequences. J. Appl. Probab. 52 (2015), no. 4, 1133--1145. doi:10.1239/jap/1450802757.

Export citation


  • Asmussen, S. and Glynn, P. W. (2007). Stochastic Simulation: Algorithms and Analysis. Springer, New York.
  • Athreya, K. B. and Roy, V. (2014). Monte Carlo methods for improper target distributions. Electron. J. Statist. 8, 2664–2692.
  • Baron, M. and Rukhin, A. L. (1999). Distribution of the number of visits of a random walk. Commun. Statist. Stoch. Models 15, 593–597.
  • Billingsley, P. (1968). Convergence of Probability Measures. John Wiley, New York.
  • Casella, G. and George, E. I. (1992). Explaining the Gibbs sampler. Amer. Statistician 46, 167–174.
  • Crane, M. A. and Iglehart, D. L. (1975). Simulating stable stochastic systems. III. Regenerative processes and discrete-event simulations. Operat. Res. 23, 33–45.
  • Durrett, R. (2010). Probability: Theory and Examples, 4th edn. Cambridge University Press.
  • Feller, W. (1971). An Introduction to Probability Theory and Its Applications, Vol. II, 2nd edn. John Wiley, New York.
  • Glynn, P. W. and Iglehart, D. L. (1987). A joint central limit theorem for the sample mean and regenerative variance estimator. Ann. Operat. Res. 8, 41–55.
  • Hobert, J. P., Jones, G. L., Presnell, B. and Rosenthal, J. S. (2002). On the applicability of regenerative simulation in Markov chain Monte Carlo. Biometrika 89, 731–743.
  • Karlsen, H. A. and Tjøstheim, D. (2001). Nonparametric estimation in null recurrent time series. Ann. Statist. 29, 372–416.
  • Kasahara, Y. (1984). Limit theorems for Lévy processes and Poisson point processes and their applications to Brownian excursions. J. Math. Kyoto Univ. 24, 521–538.
  • Metropolis, N. et al. (1953). Equations of state calculations by fast computing machines. J. Chem. Phys. 21, 1087–1092.
  • Meyn, S. P. and Tweedie, R. L. (1993). Markov Chains and Stochastic Stability. Springer, London.
  • Mykland, P., Tierney, L. and Yu, B. (1995). Regeneration in Markov chain samplers. J. Amer. Statist. Assoc. 90, 233–241.
  • Robert, C. P. and Casella, G. (2004). Monte Carlo Statistical Methods, 2nd edn. Springer, New York.
  • R Development Core Team (2011). R: A Language and Environment for Statistical Computing. R Foundation for Statistical Computing Vienna, Austria. Available at
  • Skorokhod, A. V. (1957). Limit theorems for stochastic processes with independent increments. Theory Prob. Appl. 2, 138–171.